I like this way of listing favourite rider series.

Showa: Amazon
Heisei: Ryuki
Neo-Heisei: Fourze

Since we're getting all categorical:

Showa - Black
Heisei - Ryuki
Neo Heisei - W

(I'll keep it short)
Showa: Kotaro Minami (Black/Black RX), X, and Super-1
Heisei: Kiva, Gatack, G3-X
Neo-Heisei: OOO, Knuckle, Mach

Ugh, I REALLY hate how fans have the non-existent Neo-Hesei, it's all still hesei, you don't see Toei calling it Neo-Hesei.

Showa: Amazon
Heisei: Den-O
Neo-Heisei: W

Stronger, Kuuga, Fourze. or Gaim. It's a very close tie.

To be fair, the Showa/Heisei/Neo-Heisei is a bit fan made because Shin, ZO, and J existed in the Heisei Era but are counted as Showa. It's more Who came out before Shotaro Ishinomori died and who came out after, and the Neo-Heisei era is the product of how the shows are done differently now. I mean hell you probably could have called Skyrider/Super-1/Black and Black RX Neo-Showa if you wanted because the first Seven are known as the Legendary Seven Kamen Riders. So in both the Showa and Heisei Era there are technically their own sub-categories in them. It also makes it easier to distinguish a bit.

Something that a lot of fans still haven't realized is that Kamen Rider stopped following the actual Showa/Heisei for a while now. It's really should just be called the pre-Black RX and post Kuuga generations, since that's the real split.
Neo Heisei is just a post-Decade term due to the change in format of the franchise as a whole as well as an attempt to place a greater emphasis on continuity.

Yes its all still heisei but calling it Neo-Heisei among fans is not harmful in any way shape or form. Nobody actually believes there is such thing as a Neo-Heisei era. Its a fan term to distinguish between the Pre-Decade and Post-Decade era of Heisei.

If you have a problem with the term then dont use it and dont complain just because other people are okay with using it. Nobody forces you to say it too.

And looking at how the whole Showa/Heisei thing is already royally messed up especially from the official end it doesnt matter what fans call it. If you really wanna be overly accurate just call it Kamen Rider and stop with the whole Showa/Heisei thing.

Oh, if I would sort like that, this is what it would look like:

Showa: Amazon (haven't seen any other Showa riders so far anyway)
Heisei: Blade
Neo-Heisei: OOO